>That visualization somehow escapes the demands of abstract thought. They contrast it (incorrectly) to symbolic and analytical mathematics. There is no contrast. At the root of all mathematics is abstract thought. Whether your end product is a visual proof or an analytical argument, it is the acts of abstraction and reason that are key...

You have this whole very personal ontology and theory about thinking that I don't agree too, nor does it mesh well with what can find in books, papers on the subject.

Mathematical visualization *is* thought that is *abstract*. Its not foolproof, but neither is any other kind of mathematical thought. There are no independent "demands of abstract thought" that visualization "must meet" (but that some other kind of thinking doesn't need to meet.) The "proof" criteria I mentioned earlier is not "abstract thought" itself, its a sort of public standard. How your thought operates to produce such a publicly testable result varies quite a bit, but visualization is one mode that is used by many people, but not exclusively.

What makes some kinds of visualization "mathematical" and others not? Why, intention, based on experience and training in mathematics, much of which is absorbed visually, especially in the early stages.